Pinch, punch, it’s the start of that month when the kids go back to school after the long summer break. There will be grumbling and trepidation but also many justified high hopes, thanks to so many London schools doing so well. But who do London parents turn to if and when they think something at their children’s place of learning is going wrong? And who do they think it ought to be?

A new report has found substantial variation and confusion about these things. According to research by YouGov for London Councils, parents are most likely to look to their local authority, rather than Ofsted or the Department for Education, if they have concerns about school governance or leadership, inappropriate treatment of their child by staff, educational issues or bullying.

This finding reflects many parents’ imperfect knowledge about where responsibility lies. For example, 38% of London parents with a child in an academy school believe their local authority has the power to step in if the school is under performing, and no less than 56% of those with youngsters in free schools believe the same.

In fact, local authorities have no such formal powers, as academies and free schools are run by central government. Yet many parents clearly wish they did. YouGov found strong parental support for councils having powers to intervention across the spectrum of state schools. Perhaps unsurprisingly, this was found to be highest, at 77%, among London parents with children at local authority maintained schools. However, 68% of parents with children at free schools and 63% of those with children at academies felt the same way. Even 35% of parents who’ve “gone private” would like their council to be able to step in in this way.

There was even stronger parental support for local authorities having influence over schools in their area expanding and supplying more places. At 81%, this is up by five points compared with the same time last year. London Councils thinks this particularly significant in view of its own estimate that the capital will need a further 133,000 school places, secondary and primary combined, by 2018 to accommodate its current baby boom. And it is highlighting all parts of the report suggesting parents favour local councils having more clout in education oversight and provision.

“Of course, head teachers should run schools day-to-day,” says Southwark’s leader Peter John, who is also London Councils’ executive member for children and young people. “But it’s clear from this survey that on the wider issues, parents want a council role. The government should listen to mums and dads and allow councils to act in parents’ interests.”

None of the above, though, translates directly into support or otherwise for academies and free schools. Of the YouGov sample, 31% wanted more of them, 32% didn’t and 27% were in the middle. This shows a 3% drop in support in the past year, but the broad three-way split remains. What are the lesson here? Discuss?

This article originally said London needed 133 additional school places by 2018. It should, of course, have said 133,000 additional places. This error has now been corrected.